Cauchy-Schwarz Inequality

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The Cauchy-Schwarz Inequality (or “Schwarz Inequality”)
states that for all and , we have

with equality if and only if for some scalar .

We can quickly show this for real vectors , , as follows: If either or is zero, the inequality holds (as equality). Assuming both are
nonzero, let’s scale them to unit-length by defining the normalized
vectors , , which are unit-length vectors lying
on the “unit ball” in (a hypersphere of radius ). We
have

which implies

or, removing the normalization,

The same derivation holds if is replaced by yielding

The last two equations imply

The complex case can be shown by rotating the components of
and such that becomes equal to .